Size consistency and size extensivity

In quantum chemistry, size consistency (or strict separability) is a property that guarantees the consistency of the energy behavior when interaction between the involved molecular system is nullified (for example, by distance).

Let A and B be two non-interacting systems. If a given theory for the evaluation of the energy is size consistent, then the energy of the supersystem A-B is equal to the sum of the energy of A plus the energy of B taken by themselves ($E(A-B)=E(A)+E(B)$). This property is of particular importance to obtain correctly behaving dissociation curves. Others have more recently argued that the entire potential energy surface should be well-defined.

Of the common quantum mechanical methods Hartree-Fock, coupled cluster, many-body perturbation theory (to any order), and full configuration interaction (CI) are size consistent. A major drawback of truncated CI is that it is not size-consistent and that the quality of the description decreases with increasing size of the system. The error in CISD calculations can be corrected with e.g. quadratic configuration interaction. Sometimes numerical errors can cause a method that is formally size-consistent to behave in a non-size-consistent manner.

Size-extensivity, on the other hand, is a more mathematically formal characteristic which refers to the correct (linear) scaling of a method with the number of electrons.

Core-extensivity is yet a another related property, which extends to requirement to the proper treatment of excited states.